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Whitehead groups of certain semidirect products of free groups


Author: Koo Guan Choo
Journal: Proc. Amer. Math. Soc. 43 (1974), 26-30
MSC: Primary 18F25
DOI: https://doi.org/10.1090/S0002-9939-1974-0338124-4
MathSciNet review: 0338124
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Abstract: Let $ D = {F_1} \times {F_2} \times \cdots \times {F_n}$ be a direct product of $ n$ free groups $ {F_1},{F_2}, \cdots ,{F_n},\alpha $ an automorphism of $ D$ which leaves all but one of the noncyclic factors in $ D$ pointwise fixed, $ T$ an infinite cyclic group and $ F$ another free group. Let $ D{ \times _\alpha }T$ be the semidirect product of $ D$ and $ T$ with respect to $ \alpha $ and $ (D{ \times _\alpha }T){ \times _{\alpha \times \text{id}T}}F$ the semidirect product of $ D{ \times _\alpha }T$ and $ F$ with respect to the automorphism $ \alpha \times idT$ of $ D{ \times _\alpha }T$ induced by $ \alpha $. We prove that the Whitehead group of $ (D{ \times _\alpha }T){ \times _{\alpha \times idT}}F$ and the projective class group of the integral group ring $ Z((D{ \times _\alpha }T){ \times _{\alpha \times idT}}F)$ are trivial. These results extend that of [3].


References [Enhancements On Off] (What's this?)

  • [1] H. Bass, A. Heller and R. G. Swan, The Whitehead group of a polynomial extension, Inst. Hautes Études Sci. Publ. Math. No. 22 (1964), 61-79. MR 30 #4806. MR 0174605 (30:4806)
  • [2] K. G. Choo, Whitehead groups of twisted free associative algebras, Pacific J. Math. (to appear). MR 0374175 (51:10375)
  • [3] -The projective class group of the fundamental group of a surface is trivial, Proc. Amer. Math. Soc. 40 (1973), 42-46. MR 0323869 (48:2222)
  • [4] F. T. Farrell and W. C. Hsiang, A formula for $ {K_1}{R_\alpha }[T]$, Applications for Categorical Algebra, Proc. Sympos. Pure Math., vol., 17, Amer. Math. Soc., Providence, R.I., 1970, pp. 192-218. MR 41 #5457. MR 0260836 (41:5457)
  • [5] J. Stallings, Whitehead torsion of free products, Ann. of Math. (2) 82 (1965), 354-363. MR 31 #3518. MR 0179270 (31:3518)

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DOI: https://doi.org/10.1090/S0002-9939-1974-0338124-4
Keywords: Whitehead group, projective class group, semidirect product of free groups, twisted group ring
Article copyright: © Copyright 1974 American Mathematical Society

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